Download 8 - ijssst

BABAR et al.: IMPLEMENTING DIMENSIONAL VIEW OF 4X4 LOGIC GATE
IMPLEMENTING DIMENSIONAL-VIEW OF 4X4 LOGIC
GATE/CIRCUIT FOR QUANTUM COMPUTER HARDWARE
USING XYLINX
MOHAMMAD INAYATULLAH BABAR1, SHAKEEL AHMAD3, SHEERAZ AHMED2,
IFTIKHAR AHMED KHAN1 AND BASHIR AHMAD3
1
NWFP University of Engineering & Technology, Peshawar, Pakistan
City University of Science & Information Technology, Peshawar, Pakistan
3
ICIT, Gomal University, Pakistan
2
Abstract: Theoretical constructed quantum computer (Q.C) is considered to be an earliest and foremost computing
device whose intention is to deploy formally analysed quantum information processing. To gain a computational
advantage over traditional computers, Q.C made use of specific physical implementation. The standard set of universal
reversible logic gates like CNOT, Toffoli, etc provide elementary basis for Quantum Computing. Reversible circuits
are the gates having same no. of inputs/outputs known as its width with 1-to-1 vectors of inputs/outputs mapping.
Hence vector input states can be reconstructed uniquely from output states of the vector. Control lines are used in
reversible gates to feed its reversible circuits from work bits i.e. ancilla bits. In a combinational reversible circuit, all
gates are reversible, and there is no fan-out or feedback. In this paper, we introduce implementation of 4x4
multipurpose logic circuit/gate which can perform multiple functions depending on the control inputs. The architecture
we propose was compiled in Xylinx and hence the gating diagram and its truth table was developed.
Keywords: Quantum Computer, Reversible Logic Gates, Truth Table
computations have been performed during
which operation based on quantum
computations were executed on a small no. of
qubits.
It is assumed that the Q.C will be able to
handle different problems with much faster
speed as compared to the conventional
computers, if they built on large scale [7].
The primary objects of Quantum Computing
are vectors and matrices of a Hilbert space
over the complex numbers.
Vectors are written as bras such as <Φ| and
kets such as |Ψ> . |Ψ> corresponds to a normal
(vertical) vector whereas <Φ| corresponds to
transposed (horizontal)
and
complex
conjugated vectors.
1. QUANTUM COMPUTING
Quantum computer is a computation device
that incorporates quantum mechanical
phenomena distinctively to perform operation
on data in terms of superposition &
entanglement. In contrast to traditional
computer where data is represented by means
of bits, Q.C makes use of qubits for data
representation.
Mathematical theory of
computation mainly emphasizes on to model
the computation abstraction from any
particular computer implementation.
With the development of technology,
tremendous change has been observed in
computer architectures. In addition computers
with different machine architectures can be
observed at any given time. To make them
useful mathematicians have to conceptualize
and abstract all these superficial architectural
differences away and also to focus on the
factors that really composes computation [3].
Still quantum computing known to be an
emerging
technology
and
different
IJSSST Vol. 9 No. 5, December 2008
2. BITS vs. QUBITS
Memory of conventional computer is made up
of bits and is used to hold either binary digit 1
or 0. The machine manipulates those bits by
means of transferring from logic gates to
memory and vice versa where as in Q.C,
8
ISSN 1473-804x Online, 1473-8031 Print
BABAR et al.: IMPLEMENTING DIMENSIONAL VIEW OF 4X4 LOGIC GATE
memory makes use of holding data in terms of
qubits .i.e. 1 or 0 or critically a superposition
of these. Qubits implementation for Q.C is
represented by particles having two spin states
i.e. “up” written as | 0> and “down” written as
|1 >:). They can also be entwined with other
qubits which results the astonishing
computational power of a quantum computer.
Entanglement is an exclusive quantum
observable fact. It is a property of a
multi-qubit state space and can be thought of
as a resource that the measurement on one
qubit will directly affect the other [7]. The
process of extracting information from a set of
qubits is called measurement.
Just as classical computation is based on the
ability to store and manipulate information on
collections of two-state “bits”, quantum
computation relies on ensembles of two-state
quantum systems called qubits. Unlike the
classical bit, which occupies one of two
mutually exclusive states, the qubit exists in a
superposition of two states. For our purposes,
the most general model of the qubit is:
| ψ > = α | 0 > + β |1 >
Where α and β are complex numbers such
that
| α |2 + | β |2 = 1 and
| 0 > = [ 1,0 ]T , | 1 > = [ 1,0 ]T
are orthonormal basis vectors in the
2-dimensional state space of the qubit. The
physical meaning of α and β is that any given
measurement of the qubit will show the
system to be in state | 0 >: with probability |β|2
[7]. Another way to express an arbitrary single
qubit is
| ψ > = eiγ cos
θ
| 0 > + ei (γ +φ ) sin
Figure 1: Qubit Representation
Conventional bit 1 is represented by up arrow,
bit 0 is represented by down arrow and
arrow-in-between pertains to a superposition
of 1 & 0. Moreover the arrow may be moving
around the vertical axis that pertains to the
qubit region [3].
Consider conventional computing device
operated on three bits register in which bits in
the register at any given time pertains to
definite state like 1-0-1 but in Q.C, qubits may
be in a state of all the allowed classical states.
In general, the register is described by the
following wave function:
| ψ >= a|000> + b|001 >+ c|010 >+ d |100 >+
e |110 >+ f |011 >+ g |101 >+ h|111>
Where a, b, c are coefficients for complex
numbers having probabilities expressed in
terms of amplitudes squares for measuring
each state qubits [2].
To record the register state having n qubits
quantum register (Q.R), it requires 2n complex
numbers. For instance quantum register with 3
qubits requires 23=8 numbers. It is concluded
that no. of encoded classical states in Q.R
exponentially grows with corresponding
numbers of qubits.
θ
3. THE QUANTUM CIRCUIT MODEL
|1 > [6]
2
2
Graphical representation of qubits can be
shown by sphere with arrow in it, as in the
figure below:
IJSSST Vol. 9 No. 5, December 2008
Circuits consist of wires holding different
values of bits for transferring to gates which
are responsible to perform different primitive
operations on these transferred bits.
The circuit depth is known to be total number
of time slices only in that case when the circuit
can be visualized as being divided into a
sequence of discrete time slices subject to the
9
ISSN 1473-804x Online, 1473-8031 Print
BABAR et al.: IMPLEMENTING DIMENSIONAL VIEW OF 4X4 LOGIC GATE
increase in physical entropy of process then it
is termed as physically reversible process, in
other words its known as isentropic. Moreover
no one can fix the limit to the closeness
through which one can approach perfect
reversibility. According to Rolf Landauer’s
Principal “for a computational process to be
physically reversible, it must also be logically
reversible”.
Discrete/deterministic computational process
is known as logically reversible if there is
1-to-1 mapping between old computational
states to new one and transition function.
Probably the main motivation to support
reversible computing is to enhance the
computer power consumption beyond the
primary limit of Von Neumann Landauer i.e.
kTln2 energy dissipated per irreversible bit
operation, where T is environment
temperature and the value of Bolt Mann’s
constant k=1.38 × 10-23 J/K.
application of single time slice required by a
single gate. Quantum circuitry consists of
sequence of quantum gates applied to Q.R.
Quantum register having size n is made up
with combination of uniquely addressable
qubits having individual couplings. Each
quantum gate carries reversible property
which incorporates transformation by means
of unitary operator between the inputs and
outputs. Only in that case operator U will be
known as unitary when U’U=1. Hence
quantum circuit can be defined in terms of
Q.R, to which a finite no. of quantum
operations are applied [3].
4. QUANTUM DECOHERENCE
The main problem is to keep computer
component in coherent state, because the little
interference from the external world would
effect the system to become decohere. This
may cause the computational steps of the
quantum computer to be violated. Candidate
system decoherence or de-phasing time at low
temperature usually range between seconds
and nanoseconds. In addition error correction
results in increased number of required qubits
[7].
The implementation of reversible computing
as stated in Wikipedia “Tends to typify and
control the physical dynamics of mechanisms
to carry out desired computational operations
so precisely that we can accumulate
insignificant total amount of uncertainty
regarding the complete physical state of the
mechanism per each logic operation that is
performed. Alternatively, we need to track the
state of the active energy that is involved in
carrying out computational operations within
the machine, and to design the machine in
such a way that most of this energy is
recovered in an organized form which can be
reused for subsequent operations, rather than
to dissipate into the form of heat”.
5. REVERSIBLE COMPUTATIONS AND
UNCOMPUTATION
It is currently believed that quantum
computing is the most general physical model
of computation, encompassing classical
computation. However, this is apparently
contradicted by the fact that all quantum
computations are invertible / reversible,
whereas many classical computations are
irreversible.
Any computational process which is time
invertible can be known as reversible
computing. It refers to a time-reversed version
of the process that exists within the same
general dynamical framework as the original
process. Reversibility can be of two types i.e.
physical and logical reversibility. If there is no
IJSSST Vol. 9 No. 5, December 2008
6. REVERSIBLE LOGIC GATES
Reversible logic gained much consideration in
the field of optical computing, quantum
computing and low power design.
Furthermore amalgamation of reversible logic
has become hot research area in the last decay.
In disparity to combination with conventional
10
ISSN 1473-804x Online, 1473-8031 Print
BABAR et al.: IMPLEMENTING DIMENSIONAL VIEW OF 4X4 LOGIC GATE
irreversible gates, two important limitations
for reversible gates are: a) feed back and b) fan
out are not allowed. The output comes in
terms of network consisting of cascaded
reversible gates [1].
Generally, the logic gates other than the NOT
gate in a classical computer are not reversible.
Hence, it is not possible to recover the 2 input
bits from the output bits in case of AND gate
and it is exceptional case for both input bits to
be of 1. However to describe quantum
computing device as a first step it is
enlightening to examine the theoretical
possibility of the reversible gates, moreover it
is a matter-of-fact that they are unable to
increase entropy. Reversible gates perform
reversible function on n bit data and return
corresponding n bit data, where bits string x1,
x2,…..,xn is a string having size n.
Where space {0, 1}n is formed by n-bits data
set.
Younis & Knight in [4] demonstrated that
“some reversible circuits can be made
asymptotically energy-lossless if their delay is
allowed to be arbitrarily large”. Presently
because of irreversibility, loss in energy has
been observed but this loss may be changed in
case of improvement in power dissipation [5].
For considering quantum gates, first of all we
need to specify the quantum replacement of an
n-bit datum. The quantized version of
classical n-bit space {0, 1}n is:
HBQ(n)= l2({ 0,1}n)
By classification complex valued function
space on {0, 1}n is an inner product space and
it can also be assumed as linear superposition
of bit string of classical nature. By having
notation of DirecKet, if x1 , x2 ,.........xn is
classical bit string then | x1 , x2 ,.........xn > are
known as special n qubits computational basis
states. Generally in case of Q.C we usually
take interest in reversible gates but here we
shall concentrate in unitary mapping i.e.
mainly used to preserve inner product on
HQB(n). Actually quantum gates are offered
rise by classical reversible n-bit logic gates in
IJSSST Vol. 9 No. 5, December 2008
terms of: for every reversible n-bit logic gate
“f” the corresponding quantum gate “Wf” may
be described as Wf (| x1, x2,……..,xn >)=|f (x1,
x2,……..,xn)>.
1 qubit gate represented by relative phase shift
can be described by the unitary matrix:
⎛ etθ
Uθ = ⎜
⎝ 0
so
0⎞
⎟
1⎠
U θ |0>=et θ |0> U θ |1>=|1>
7. QUANTUM GATE
NOR, AND, NAND and XOR are all
irreversible logic gates; they all must generate
heat. Amount of information on the right hand
side of below given equation is less than the
information on the left hand side [3].
(a,b)→- (a ^ b)
Special gates have been conceived and
fabricated which maintain all information that
is passed to them, so that the computation can
be run forward and backward. The
computational results in a very large amount
of junk, because every intermediate step is
remembered, but heat generation is eliminated
while the computation goes on. Quantum
circuits can be described by quantum logic
gates which may perform operation on small
number of qubits. They can be compared with
the conventional logic gates of digital
computers. In contrast to conventional logic
gates these are reversible. For example Toffoli,
a universal logic gate can be directly mapped
to quantum logic gates and it also provides
reversibility. Quantum logic gates can also be
represented by unitary matrices [7].
8. LOGIC IMPLEMENTATION OF 4x4
PROPOSED CIRCUIT MODEL
The logic diagram given below in figure 4
represents the 4x4 circuit model implemented
11
ISSN 1473-804x Online, 1473-8031 Print
BABAR et al.: IMPLEMENTING DIMENSIONAL VIEW OF 4X4 LOGIC GATE
in Xylinx which has a,b,c, and d acting as
inputs and w,x,y and z are the outputs
producing results on the basis of the input
combinations and the pseudo-code for the
implementation is given in the next section.
Figure 4: 4x4 Multipurpose Reversible Logic Gate/Circuit
output reg w,x,y,z;
always@(*)
begin
if (c==0 && d==0)
begin
w=a&b; x=a|b; z=a&(~b);
end
if (c==0 && d==1)
begin
w=a|b; x=a^b; y=a&(~b); z=~a;
end
if (c==1 && d==0)
begin
w=a&b; x=a^b; y=~(a&b);
end
9. PSEUDOCODE FOR THE PROPOSED
LOGIC CIRCUIT
As regards the logic circuit proposed, the
inputs to the logic circuit are a,b,c,d whereas
the outputs are w,x,y,z. a and b are acting as
control inputs whereas c and d are acting as
normal inputs. Depending on the different
combinations of the control inputs and for
different combination of the normal inputs, it
performs different logical operations
according to the following logical
operations.[7]
module quantum_gate(w,x,y,z,a,b,c,d);
input a,b,c,d;
IJSSST Vol. 9 No. 5, December 2008
12
ISSN 1473-804x Online, 1473-8031 Print
BABAR et al.: IMPLEMENTING DIMENSIONAL VIEW OF 4X4 LOGIC GATE
if (c==1 && d==1)
begin
w=a|b; x=a&b; y=a&(~b);
end
1
11.
The following table 1 gives the truth table for
the model circuit where the inputs are
designated as a,b,c,d and outputs are
designated as w,x,y,z. a and b are acting as
control inputs whereas c and d are acting as
normal inputs. As there are 4 inputs so there
can be a maximum of 16 combinations of 0s
and 1s. Depending on the different
combinations we have different outputs where
0s represent low-level states and 1s represent
high-level states and Xs represent the don’t
care conditions or the garbage outputs. These
outputs have been formulated according to the
Boolean logical expressions giving different
input combinations [7].
B
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
a
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
w
0
0
0
1
0
0
0
1
0
1
1
1
0
1
1
X
0
1
1
1
0
1
1
0
0
1
1
0
0
0
0
IJSSST Vol. 9 No. 5, December 2008
y
X
X
X
X
1
1
1
0
0
1
0
0
0
1
0
1
1
1
0
X
CONCLUSIONS
WORK
AND
FUTURE
Many applications in vast area of
telecommunication systems, cryptography
and computer graphics are related to
reversible or information lossless circuits.
Also they become increasingly important for
the emerging field of Quantum Computing.
The aim of this research project is focused on
the improvement of hardware circuitry for
Quantum Computing by designing a novel
circuit which can be regarded as a reversible
logic circuit. The circuit shall be designed as a
model so that further improvements in this
regard are possible in the future. The existing
non-reversible logic gates dissipate a
considerable amount of energy and their
combinational circuits even more. The circuit
model thus suggested can be utilized in energy
and power efficient digital circuits as
compared to the already existing power
dissipation digital circuits. The current design
does not develop XNOR and NOR functions
but it can be extended and modified to
perform these desired operations. The model
thus proposed does not confirm to the
characteristic of reversibility as per the
requirement of quantum circuits or gates and
in the future work we shall focus on its
improvement to be functioning as reversible
as well.
10. TRUTH TABLE FOR THE MODEL
CIRCUIT
c
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
Table 1: Truth Table for Proposed Gate
end
endmodule
D
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
Z
0
1
0
0
X
X
X
X
1
0
1
0
X
X
X
REFERENCES
1. Daniel GroBe, Xiaobo Chen, Gerhard
W. Dueck, Rolf Drechsler , Exact
SAT-based
Toffoli
Network
Synthesis , Department of Computer
Science, , Stresa-Lago Maggiore, Italy
GLSVLSI’07, March 11–13, 2007.
2. Zdzislaw Meglicki, “Introduction to
Quantum Computing”, February 5,
13
ISSN 1473-804x Online, 1473-8031 Print
BABAR et al.: IMPLEMENTING DIMENSIONAL VIEW OF 4X4 LOGIC GATE
3.
4.
5.
6.
7.
2002.
Alexei Kitaev, William A.Webb,
“Wavefunction Preparation using a
Quantum Computer”, airXiv: IEEE
Trans. on CAD, vol 22(6), June 2003,
p. 714-726 (preliminary version in
Proc. ICCAD 2002, pp.353-360).
S.Younis , T.Knight, “Asymptotically
Zero Energy Split-Level Charge
Recovery Logic”, Workshop on Low
Power Design, 1994.
Vivek V. Shinde, Aditya K. Prasad,
Igor L. Markov, John P. Hayes,
“Reversible Logic Circuit Synthesis”,
IEEE, 2002.
Mark
Oskin,
“Quantum
Computing-Lecture
Notes”,
Department of Computer Science and
engineering,
University
of
washington.
Sheeraz Ahmed, Abdus Salaam, “4x4
Multipurpose Logic Gate/Circuit for
Quantum Hardware, Realizing on the
Same Outputs”, ICICT 2008, USTB,
Pakistan.
IJSSST Vol. 9 No. 5, December 2008
14
ISSN 1473-804x Online, 1473-8031 Print